Electron. J. Diff. Eqns., Vol. 2003(2003), No. 110, pp. 1-4.

A remark on the existence of large solutions via sub and supersolutions

Jorge Garcia-Melian

We study the boundary blow-up elliptic problem $\Delta u=a(x) f(u)$ in a smooth bounded domain $\Omega\subset \mathbb{R}^N$, with $u|_{\partial\Omega}=+\infty$. Under suitable growth assumptions on $a$ near $\partial\Omega$ and on $f$ both at zero and at infinity, we prove the existence of at least a positive solution. Our proof is based on the method of sub and supersolutions, which permits on the one hand oscillatory behaviour of $f(u)$ at infinity and on the other hand positive weights $a(x)$ which are unbounded and/or oscillatory near the boundary.

Submitted July 4, 2003. Published November 4, 2003.
Math Subject Classifications: 35J60, 35J25.
Key Words: Boundary blow-up, sub and supersolutions

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Jorge Garcia-Melian
Dpto. de Analisis Matematico, Universidad de La Laguna
c. Astrofisico Francisco Sanchez s/n, 38271 - La Laguna, Spain

Centro de Modelamiento Matematico, Universidad de Chile
Blanco Encalada 2120, 7 piso - Santiago, Chile
email: jjgarmel@ull.es

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